If your definition of pure math is "proving theorems", she can work in the following (math related to the following): 1). Control theory (this can mean doing dynamical systems, PDEs, etc): PhDs hired by several leading companies such as Ford, Bosch, Honda, United Technologies 2). Information theory and networks : Internet companies 3). Optimization and/or nonlinear programming: PhDs hired by numerous companies 4). Communication theory: PhDs hired by communication giants 5). Stochastic geometry: Same as 4 6). Probability: Finance/banks/internet